Problem: Find the quadratic polynomial, with real coefficients, which has $3 + i$ as a root, and where the coefficient of $x^2$ is 2.
Solution: Since the polynomial has real coefficients, the other root must be $3 - i.$  Thus, the polynomial is
\begin{align*}
2(x - 3 - i)(x - 3 + i) &= 2((x - 3)^2 - i^2) \\
&= 2((x - 3)^2 + 1) \\
&= \boxed{2x^2 - 12x + 20}.
\end{align*}